Polar Graphs

1
Polar Graphs

• 8.2

2
Converting From Polar to Cartesian

• Let the point P have polar coordinates (r, ?) and
  rectangular coordinates (x,y). Then
• x r cos ? r2 x2 y2
• y r sin ? tan ? y/x

3
Polar Graphing on the Calculator

• Select the mode button and choose Pol.
• The put the equation in y, which is r1.
• Try zoom 6 and then press the window key to
  change ?min, ?max, and ?Step.
• ?Step just changes the number of calculations the
  calculator makes for plots.

4
Horizontal Line--r sin ? a

• Is a horizontal line a units above the pole or
  origin if agt0 and a units below the pole if
  alt0.
• Assume that a is not zero.

5
Vertical Line--r cos ? a

• Is a vertical line a units to the right of the
  pole or origin if a is positive and a units to
  the left of the pole if a is negative.

6
Line--? ?o

• The line which makes an angle ?o of with the
  x-axis. The angle is fixed at a given value, but
  the radius can be anything, so this creates a
  line that creates the fixed angle with the x-axis.

7
Circle--ra

• Circle of radius a centered at the origin. The
  radius is fixed at a, but the angle can be
  anything.

8
Circle--r /-2a sin ?

• Circle with radius a and passes through the pole.
• Sine will orient along the y-axis
• Centered at (0,/-a) in rectangular coordinates

9
Circle--r /-2a cos ?

• Circle with radius a and passes through the pole.
• Cosine will orient on the x-axis.
• In rectangular coordinates, the center will be at
  (/-a,0)

10
Archimedes Spiral--Ra?

• The angle varies as the radius varies. In fact,
  as the angle gets larger, so does the radius.

11
Rose Curves

• r a cos n? or r a sin n?

12
Cardiods

• ra(1/- cos ?)
• Or
• ra(1 /- sin ?)

13
Limacons

• R a /- b cos ?
• R a /- b sin ?
• If agt0, and bgt0, and agtb, then there is no inner
  loop.
• If agt0, and bgt0, and altb, then there is an inner
  loop.

14
Lemniscates

• r2 a2sin(2?) or
• r2 a2cos(2?)

15
Converting Polar Equations to Rectangular

• Use the conversion equations.
• Note that r sqrt(x2y2)
• Perhaps, multiply both sides by r, or top and
  bottom of a fraction by r so that you can use one
  of the conversion equations.

16
Symmetry

• With Respect to the Polar Axis
• In the polar equation, replace ? with ?. If an
  equivalent equation results, the graph is
  symmetric with respect to the polar axis.
• With Respect to the Line ? p/2
• In a polar equation, replace ? with (p ?). If
  an equivalent equation results, the graph is
  symmetric with respect to the polar axis.
• With Respect to the Pole
• In a polar equation, replace r with r. If an
  equivalent equation results, then graph is
  symmetric with respect to the pole.